Order domain analysis of speed-dependent friction-induced torque in a brake experiment

A friction-induced forced vibration problem, as excited by the geometric distortions of the brake rotor, is studied in this article. The focus is on the order domain analysis, as the speed-dependent behavior of friction torque is not well understood. First, a new laboratory experiment is constructed to simulate vehicle brake judder in a scientific and yet controlled manner. The variations in pressure and torque are measured as the rotor slows down, and the order domain tracking is used to construct shaft torque vs. speed diagrams. A quasi-linear model of the laboratory experiment is then developed to obtain an analytical solution and to estimate the torque envelope function. A nonlinear model of the laboratory experiment (with a clearance) is also investigated to examine the resonant amplitude growth. Finally, predictions are successfully compared with measurements. Several contributions emerge over the prior literature. In particular, the experimental data clearly show that multiple-orders of the rotor surface distortion profile excite the friction-induced torque, and a clearance in the torsional system controls the resonant amplitude regime. New analytical and numerical solutions provide much insight into the speed-dependent resonant amplitude growth process.     

Dynamics of a 3dof torsional system with a dry friction controlled path

A three-degrees of freedom semi-definite torsional system representing an automotive driveline is studied in presence of a torque converter clutch that manifests itself as a dry friction path. An analytical procedure based on the linear system theory is proposed first to establish the stick-to-slip boundaries. Smoothened and discontinuous Coulomb friction formulations are then applied to the nonlinear system, and the differential governing equations are numerically solved given harmonic torque excitation and a mean load. Time domain histories illustrating dry friction-induced stick–slip motions are predicted for different saturation torques and system parameters. Approximate analytical solutions based on distinct states are also developed and successfully compared with numerical studies. Analysis shows that the conditioning factor associated with the smoothened friction model (hyperbolic tangent) must be carefully selected. Then nonlinear frequency responses are constructed from cyclic time histories and the stick–slip boundaries predictions (as yielded by the linear system theory) are confirmed. In particular, the effect of secondary inertia is analytically and numerically investigated. Results show that the secondary inertia has a significant influence on the dynamic response. A quasi-discontinuous oscillation is found with the conventional bi-linear friction model in which the secondary inertia is ignored. Finally, our methods are successfully compared with two benchmark analytical and experimental studies, as available in the literature on two-degrees of freedom translational systems.     

Influence of harmonically varying normal load on steady-state behavior of a 2dof torsional system with dry friction

A nonlinear dry friction problem with harmonically varying normal load is formulated, in the context of a two-degree of freedom torsional system, since virtually all of the prior literature focuses on the topic of time-invariant normal load. First, pure stick, pure slip and stick-slip motions are computationally and analytically determined when excited by a sinusoidal torque, in the presence of harmonically varying saturation torque; mean terms are included in both. These analyses yield both transient and steady-state time histories under various conditions. Second, the effects of time-varying normal load on steady-state responses have been investigated and nonlinear spectral maps (including super-harmonics) are developed. Results show that the actuation system parameters could affect steady-state stick-slip motions in different ways over the lower and higher frequency regimes, as a result of time-delay in slip motions with respect to the torque excitation. In particular, the negative slope characteristics in the friction law exaggerate the stick-slip vibration problems, and it is the major cause of bifurcations and quasi-periodic or chaotic motions. Around the super-harmonic peak frequencies, the nonlinear system tends to lose stability as abrupt jumps in the spectral maps take place. An equivalent viscous damping model is considered to analytically investigate the instability mechanism. Further, the periodicity of the system response under harmonically varying actuation is conceptually by employing the harmonic balance method. Finally, steady-state behavior is examined for the nonlinear, time-varying dry friction problem.     

Optical actuation of a macroscopic mechanical oscillator

An intensity-modulated HeNe-laser beam was utilized to optically actuate the mechanical resonance of a macroscopic torsional silicon oscillator (f₀ = 67 700 Hz, Q = 42 100 at p = 1 mbar and T = 300 K). Both radiation pressure and photothermal effects may cause optical actuation of a mechanical device. Both excitation effects were studied. In actuation through radiation pressure, the actuating laser beam was focused on the high-reflectivity-coated oscillator surface. In the case where the intensity-modulated laser beam was incident on the uncoated silicon surface the photothermal effect was shown to be the dominating excitation factor. Oscillation amplitudes due to the actuation through radiation pressure and photothermal effects were Δ x_rad = 1.4 pm and Δ x_ph = 4.3 pm with the same optical power of 1.5 mW. The measured resonance frequency and quality value were not changed when purely mechanical and radiation pressure actuation mechanisms were compared. With photothermal actuation the absorbed optical power heats the oscillator, introducing a slight decrease in the resonance frequency. Our experiments demonstrate that optical actuation combined with sensitive optical interferometric measurements can be utilized to perform dynamic vibration analysis of micromechanical components. Prospects of using micromechanical devices for observing extremely weak external forces are discussed.     

柴油机轴系扭振系统的强非线性主共振

研究柴油机轴系扭振系统强非线性问题.根据拉格朗日方程建立柴油机轴系扭振系统的动力学模型,通过参数变换,应用Modified Lindstedt-Poincaré方法得到柴油机轴系扭振系统强非线性主共振的幅频响应方程,分析系统不同参数对主共振幅频响应的影响.结果表明,系统的幅频响应曲线存在跳跃,随着简谐力矩的减小和阻尼的增大,系统的非线性跳跃减弱,系统的振幅减小,系统主共振的区域也随之减小;随着调谐参数的变化,系统的主共振力幅响应曲线存在两种拓扑结构.MLP方法得出的近似解析解与龙格库塔法得出的数值解吻合.     

Modeling of automotive drum brakes for squeal and parameter sensitivity analysis

Many fundamental studies have been conducted to explain the occurrence of squeal in disc and drum brake systems. The elimination of brake squeal, however, still remains a challenging area of research. Here, a numerical modeling approach is developed for investigating the onset of squeal in a drum brake system. The brake system model is based on the modal information extracted from finite element models for individual brake components. The component models of drum and shoes are coupled by the shoe lining material which is modeled as springs located at the centroids of discretized drum and shoe interface elements. The developed multi degree of freedom coupled brake system model is a linear non-self-adjoint system. Its vibrational characteristics are determined by a complex eigenvalue analysis. The study shows that both the frequency separation between two system modes due to static coupling and their associated mode shapes play an important role in mode merging. Mode merging and veering are identified as two important features of modes exhibiting strong interactions, and those modes are likely candidates that lead to coupled-mode instability. Techniques are developed for a parameter sensitivity analysis with respect to lining stiffness and the stiffness of the brake actuation system. The influence of lining friction coefficient on the propensity to squeal is also discussed.     

Free torsional vibration of nanotubes based on nonlocal stress theory

A new elastic nonlocal stress model and analytical solutions are developed for torsional dynamic behaviors of circular nanorods/nanotubes. Unlike the previous approaches which directly substitute the nonlocal stress into the equations of motion, this new model begins with the derivation of strain energy using the nonlocal stress and by considering the nonlinear history of straining. The variational principle is applied to derive an infinite-order differential nonlocal equation of motion and the corresponding higher-order boundary conditions which contain a nonlocal nanoscale parameter. Subsequently, free torsional vibration of nanorods/nanotubes and axially moving nanorods/nanotubes are investigated in detail. Unlike the previous conclusions of reduced vibration frequency, the solutions indicate that natural frequency for free torsional vibration increases with increasing nonlocal nanoscale. Furthermore, the critical speed for torsional vibration of axially moving nanorods/nanotubes is derived and it is concluded that this critical speed is significantly influenced by the nonlocal nanoscale.     

高效半解析方法分析周期正交加筋板的振动-声辐射特性

为了对水下无穷大双周期正交加筋板结构模型在简谐面力激励下的振动响应及声辐射特性进行更为合理的理论预测与分析,建立了加筋板结构的数学模型。结合傅里叶变换、泊松迭加公式及空间波数法,将周期加筋板的振动响应及辐射声压表达为关于结构位移谐波分量的函数方程,对加筋板模型提出了高效分析求解方法并进行了谐波分量截断求解。验证了方法的正确性,并分析了结构的振动特性以及加强筋周期间距和扭矩对辐射声压的影响。结果表明,加强筋的扭转作用影响加筋板结构的振动模态频率,对于较高精度要求的工程应用,加强筋的扭转作用不能忽略。通过调节加强筋周期间距及横截面尺寸,可以降低薄板在较低频域区间的远场辐射声压。      

External and internal coupling effects of rotor's bending and torsional vibrations under unbalances

External and internal bending–torsion coupling effects of a rotor system with comprehensive unbalances are studied by analytical analysis and numerical simulations. Based on Lagrangian approach, a full-degree-of-freedom dynamic model of a Jeffcott rotor is developed. The harmonic balance method and the Floquet theory are combined to analyze the stability of the system equations. Numerical simulations are conducted to observe the bending–torsion coupling effects. In the formulation of rotordynamic model, two bending–torsion coupling patterns, external coupling and internal coupling, are suggested. By analytical analysis, it is concluded that the periodic solution of the system is asymptotically stable. From numerical simulations, three bending–torsion coupling effects are observed in three cases. Under static unbalance, synchronous torsional response is observed, which is the result of external coupling under unbalanced force. Under dynamic unbalance, two-time synchronous frequency torsional response is observed, which is the result of internal coupling under unbalanced moment. Under comprehensive unbalance, synchronous and two-time synchronous frequency torsional components are observed, which are the results of both external and internal couplings under unbalanced force and moment. These observations agree with the analytical analysis. It is believed that these observed phenomena should make sense in the dynamical design and fault diagnostics of a rotor system.     

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.     

Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

Dark spatial solitons in discrete cubic media with self- and cross-phase modulation

We analytically study the background stability of dark strongly localized vectorial modes in discrete cubic media with self- and cross-phase modulation. The instability regions are identified as a function of the linear and non-linear coupling strength as well as the ratio between the background amplitudes of the two components. The respective instability gain is calculated. Approximate analytical solutions for vectorial discrete solitons of different topologies are derived. The analytical results obtained are confirmed by direct numerical simulations.     

纳秒脉冲表面介质阻挡等离子体激励唯象学仿真

结合NS-DBD实验数据和理论分析, 建立NS-DBD单区非均匀唯象学模型, 旨在通过合理的模型进行流动控制仿真, 揭示流动控制机理. 在平板无来流时, 运用单区非均匀唯象学模型, 通过引入涡量输运方程, 求解涡量方程各项, 分析展向涡形成机理. 展向涡主要是由压力升诱导激励区压力梯度和密度梯度的不正交性产生的, 其次是激励区附近流场的对流引起的涡量转移. 圆柱上的激励仿真得到与实验一致的压缩波结构和冲击波位置, 验证了模型合理性. NACA 0015翼型大迎角分离控制的仿真表明, 激励诱导展向涡促使主流和分离流相互作用, 使分离点移向下游; 脉冲激励频率通过诱导展向涡的数量对流动分离产生不同的作用效果, 本文最佳的无量纲激励频率为6.     

Energy flow model for thin plate considering fluid loading with mean flow

Energy Flow Analysis (EFA) has been developed to predict the vibration energy density of system structures in the high frequency range. This paper develops the energy flow model for the thin plate in contact with mean flow. The pressure generated by mean flow affects energy governing equation and power reflection–transmission coefficients between plates. The fluid pressure is evaluated by using velocity potential and Bernoulli's equation, and energy governing equations are derived by considering the flexural wavenumbers of a plate, which are different along the direction of flexural wave and mean flow. The derived energy governing equation is composed of two kinds of group velocities. To verify the developed energy flow model, various numerical analyses are performed for a simple plate and a coupled plate for several excitation frequencies. The EFA results are compared with the analytical solutions, and correlations between the EFA results and the analytical solutions are verified.     

Interaction of fundamental parametric resonances with subharmonic resonances of order one-half

The interaction of fundamental parametric resonances with subharmonic resonances of order one-half in a single-degree-of-freedom system with quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to derive two first-order ordinary differential equations that describe the modulation of the amplitude and the phase of the response with the non-linearity and both resonances. These equations are used to determine the steady state solutions and their stability. Conditions are derived for the quenching or enhancement of a parametric resonance by the addition of a subharmonic resonance of order one-half. The degree of quenching or enhancement depends on the relative amplitudes and phases of the excitations. The analytical results are verified by numerically integrating the original governing differential equation.     

Determination of the dynamic elastic moduli and internal friction using thin resonant bars

Analysis of the data from the resonant-bar technique for determining wave speed and internal friction is presented. Internal friction has been included in the longitudinal and torsional wave equations and the analytical solution has been obtained. In determining the acoustical constants of lossy materials, a broad frequency spectrum is used that includes many resonances and not just data at or near the individual resonances. Corrections due to the mass, length, stiffness, and damping of the transducers are also presented. The theoretical solutions are compared with the measured magnitude and phase response data for torsional, longitudinal, and flexural measurements and are shown to be in good agreement.     

Analytical Study on Propagation Dynamics of Optical Beam in Parity-Time Symmetric Optical Couplers

We present exact analytical solutions to parity-time(P T) symmetric optical system describing light transport in P T-symmetric optical couplers. We show that light intensity oscillates periodically between two waveguides for unbroken P T-symmetric phase, whereas light always leaves the system from the waveguide experiencing gain when light is initially input at either waveguide experiencing gain or waveguide experiencing loss for broken P T-symmetric phase. These analytical results agree with the recent experimental observation reported by Ru¨ter et al. [Nat. Phys.6(2010) 192]. Besides, we present a scheme for manipulating P T symmetry by applying a periodic modulation. Our results provide an efficient way to control light propagation in periodically modulated P T-symmetric system by tuning the modulation amplitude and frequency.     

Exploring the dynamics of a class of post-tensioned, moment resisting frames

In this paper, we present an equivalent low-order nonlinear system that describes the dynamics of a generic class of post-tensioned frames. The proposed nonlinear single degree of freedom system is derived from energy considerations. We demonstrate that the equation of motion for the entire, planar, post-tensioned frame is equivalent to the dynamics of a single tied rocking block on an elastic foundation. As validation for this analytical model we present physical tests (1/4 scale) undertaken at Bristol. Quasi-static push-pull-over tests and dynamic frequency sine sweep shake table tests are conducted on the physical model. Comparison of results indicate that the analytical model predicts both quasi-static nonlinear push-over and nonlinear dynamic resonant behaviour very well. Further numerical simulations on the analytical model identify the nonlinear resonant frequency backbone curves for a range of system parameters. We explore catchment basins of both Poincaré phase and system parameter spaces. In addition we describe failure boundaries and system integrity surfaces giving an indication as to likely bounds on forcing amplitudes.